Sum of Beauty in the Array

For the standard version, an interior value earns two points if every value to its left is smaller and every value to its right is larger; otherwise it earns one point if it is larger than its left neighbor and smaller than its right neighbor. Precompute prefixMax[i], the maximum through index i, and suffixMin[i], the minimum from index i onward. For each i from 1 to n-2, first test prefixMax[i-1] < nums[i] < suffixMin[i+1]; if true, add two. Otherwise test the local relation nums[i-1] < nums[i] < nums[i+1] and add one if true. Strict comparisons matter because duplicates do not satisfy the definition. Precomputation and the final scan take O(n) time and O(n) space, or O(n) auxiliary space with a suffix array and running prefix maximum. Do not confuse local sortedness with the stronger global condition.